Stability of the periodic Toda lattice under short range perturbations

Abstract

We consider the stability of the periodic Toda lattice (and slightly more generally of the algebro-geometric finite-gap lattice) under a short range perturbation. We prove that the perturbed lattice asymptotically approaches a modulated lattice. More precisely, let $g$ be the genus of the hyperelliptic curve associated with the unperturbed solution. We show that, apart from the phenomenon of the solitons travelling on the quasi-periodic background, the $n/t$-pane contains $g+2$ areas where the perturbed solution is close to a finite-gap solution in the same isospectral torus. In between there are $g+1$ regions where the perturbed solution is asymptotically close to a modulated lattice which undergoes a continuous phase transition (in the Jacobian variety) and which interpolates between these isospectral solutions. In the special case of the free lattice ($g=0$) the isospectral torus consists of just one point and we recover the known result. Both the solutions in the isospectral torus and the phase transition are explicitly characterized in terms of Abelian integrals on the underlying hyperelliptic curve. Our method relies on the equivalence of the inverse spectral problem to a matrix Riemann--Hilbert problem defined on the hyperelliptic curve and generalizes the so-called nonlinear stationary phase/steepest descent method for Riemann--Hilbert problem deformations to Riemann surfaces.

38 pages, 1 figure. This version combines both the original version and arXiv:0805.3847

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Cited by in corpus (11)

• Growth rate of modulation instability driven by superregular breathers
• Soliton Solutions of the Toda Hierarchy on Quasi-Periodic Backgrounds Revisited
• On the Dirichlet to Neumann Problem for the 1-dimensional Cubic NLS equation on the Half-Line; Zero Initial Data
• Wave phenomena of the Toda lattice with steplike initial data
• Long-Time Asymptotics for the Toda Shock Problem: Non-Overlapping Spectra
• Trace Formulas for Schroedinger Operators in Connection with Scattering Theory for Finite-Gap Backgrounds
• Stability of the Periodic Toda Lattice: Higher Order Asymptotics
• Scattering theory with finite-gap backgrounds: Transformation operators and characteristic properties of scattering data
• Nonlinear steepest descent on a torus: A case study of the Landau-Lifshitz equation
• The Focusing NLS Equation on the Half-Line with Periodic Boundary Conditions
• A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions